Theorem uniin

Description: The class union of the intersection of two classes. Exercise 4.12(n) of Mendelson p. 235. See uniinqs for a condition where equality holds. (Contributed by NM, 4-Dec-2003) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Assertion	uniin	⊢ ∪ ( 𝐴 ∩ 𝐵 ) ⊆ ( ∪ 𝐴 ∩ ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		19.40	⊢ ( ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) → ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
2		elin	⊢ ( 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) )
3	2	anbi2i	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝑦 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) )
4		anandi	⊢ ( ( 𝑥 ∈ 𝑦 ∧ ( 𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
5	3 4	bitri	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
6	5	exbii	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) ↔ ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
7		eluni	⊢ ( 𝑥 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) )
8		eluni	⊢ ( 𝑥 ∈ ∪ 𝐵 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) )
9	7 8	anbi12i	⊢ ( ( 𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵 ) ↔ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∧ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
10	1 6 9	3imtr4i	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) → ( 𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵 ) )
11		eluni	⊢ ( 𝑥 ∈ ∪ ( 𝐴 ∩ 𝐵 ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∩ 𝐵 ) ) )
12		elin	⊢ ( 𝑥 ∈ ( ∪ 𝐴 ∩ ∪ 𝐵 ) ↔ ( 𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵 ) )
13	10 11 12	3imtr4i	⊢ ( 𝑥 ∈ ∪ ( 𝐴 ∩ 𝐵 ) → 𝑥 ∈ ( ∪ 𝐴 ∩ ∪ 𝐵 ) )
14	13	ssriv	⊢ ∪ ( 𝐴 ∩ 𝐵 ) ⊆ ( ∪ 𝐴 ∩ ∪ 𝐵 )